Rainbow AP(4) in an almost equinumerous coloring

 Importance: Medium ✭✭
 Author(s): Conlon, David
 Subject: Combinatorics
 Keywords: arithmetic progression rainbow
 Posted by: vjungic on: July 20th, 2007

\begin{problem} Do 4-colorings of $\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either $\lfloor p/4\rfloor$ or $\lceil p/4\rceil$? \end{problem}

It is known that there are equinumerous colorings of $\mathbb{Z}_{4m}$ (i.e. colorings of $\mathbb{Z}_{4m}$ for some $m$ such that each color occurs $m$ times) within which we cannot find rainbow arithmetic progressions of length $4$. (\cite{CJR})

Bibliography

% Example: *[C] David Conlon, Rainbow solutions of linear equations over $\mathbb{Z}_p$, Discrete Mathematics, 306 (2006) 2056 - 2063.

[CJR] David Conlon, Veselin Jungic, Rados Radoicic, \href[On the existence of rainbow 4-term arithmetic progressions]{http://dx.doi.org/10.1007/s00373-007-0723-2}, Graphs and Combinatorics, 23 (2007), 249-254

* indicates original appearance(s) of problem.

Tight hypergraphs

It deservs to be mentioned that in any $3$-colouring of ${\mathbb Z}_p^*/{\mathbb Z}_3^*$, the equation $x+y=z$, have an heterochromatic (rainbow) solution; here, ${\mathbb Z}_p^*$ denotes the multiplicative group of the field ${\mathbb Z}_p$.